manipulation and steering of hyperbolic surface polaritons in...

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Manipulation and Steering of Hyperbolic Surface Polaritons in Hexagonal Boron Nitride

Siyuan Dai, Mykhailo Tymchenko, Yafang Yang, Qiong Ma, Marta Pita-Vidal, Kenji Watanabe, Takashi Taniguchi, Pablo Jarillo-Herrero, Michael M. Fogler, Andrea Alù,* and Dimitri N. Basov*

Dr. S. Dai, Prof. M. M. Fogler, Prof. D. N. BasovDepartment of PhysicsUniversity of CaliforniaSan Diego, La Jolla, CA 92093, USAE-mail: [email protected]. S. Dai, M. Tymchenko, Prof. A. AlùDepartment of Electrical and Computer EngineeringThe University of Texas at AustinAustin, TX 78712, USAE-mail: [email protected]. Yang, Dr. Q. Ma, M. Pita-Vidal, Prof. P. Jarillo-HerreroDepartment of PhysicsMassachusetts Institute of TechnologyCambridge, MA 02215, USADr. K. Watanabe, Dr. T. TaniguchiNational Institute for Materials ScienceNamiki 1-1, Tsukuba, Ibaraki 305-0044, JapanProf. D. N. BasovDepartment of PhysicsColumbia UniversityNew York, NY 10027, USA

The ORCID identification number(s) for the author(s) of this article can be found under https://doi.org/10.1002/adma.201706358.

DOI: 10.1002/adma.201706358

and recently discovered experimentally in a range of natural materials.[11–20] Many natural hyperbolic materials belong to the class of van der Waals (vdW) crystals,[21] highly anisotropic compounds in which atomic layers are bonded together through vdW forces.[22] Hexagonal boron nitride (hBN) is a prototypical vdW material natu-rally possessing both Type I (εxy > 0, εz < 0 where εxy and εz are permittivities in the basal plane and along the optical axis) and Type II (εxy < 0, εz > 0) 3D hyperbolicity that stems from its anisotropic phonon resonances.[11,12,23,24] Recent works have

reported on the nanoimaging of propagating polaritons–elec-tromagnetic waves hybridized between photons and dipolar modes[21,25]–in hyperbolic systems. In hBN, both volumetric hyperbolic polaritons (HPs)[11,12,14,15,23,26–29] and surface-confined hyperbolic surface polaritons (HSPs)[20] have been observed. HPs and HSPs have been shown to possess signifi-cantly improved polaritonic loss and upper momentum cut-off due to the hBN’s natural lattice.[26,27] Therefore, HPs and HSPs in hBN are superior to those in metamaterials and metasur-faces that inevitably suffer from finite feature sizes of constit-uent elements and fabrication imperfections. While previous works focused on the intrinsic properties of HSPs;[18,20] here, we combine infrared (IR) nanoimaging and finite element simulations to demonstrate the manipulation of reflection, transmission, scattering, and propagation steering of HSPs in tailored hBN nanostructures (see Experimental Section).

Infrared nanoimaging experiments were performed in the Type II hyperbolic region of hBN using a scattering-type scan-ning near-field optical microscope (s-SNOM) (see Experimental Section).[11,23,27] A metalized atomic force microscope (AFM) tip was illuminated by IR light (Figure 1a), generating strong elec-tromagnetic near fields underneath the apex. These fields excite a wide range of momenta q over the surface and therefore facil-itate the formation of coupled photon-lattice modes referred to as phonon polaritons. In the experiment, the AFM tip acts both as a launcher[30] and a detector of polaritons supported in hBN. Standing wave oscillations of the propagating polaritons are recorded as samples and are scanned underneath the AFM tip.[11,15,23,27] The s-SNOM images of hBN nanostructures at a representative IR frequency ω = 1425 cm−1 (ω = 1/λIR where λIR is the IR wavelength) are presented in Figure 1. On the top surface of a hBN cuboid (Figure 1a), we observe a series of oscillations of nano-IR signal that we represent in the form of

Hexagonal boron nitride (hBN) is a natural hyperbolic material that supports both volume-confined hyperbolic polaritons and sidewall-confined hyperbolic surface polaritons (HSPs). In this work, efficient excitation, control, and steering of HSPs are demonstrated in hBN through engineering the geometry and orientation of hBN sidewalls. By combining infrared nanoimaging and numerical simulations, the reflection, transmission, and scattering of HSPs are investigated at the hBN corners with various apex angles. It is also shown that the sidewall-confined nature of HSPs enables a high degree of control over their propagation by designing the geometry of hBN nanostructures.

Surface Polaritons

Hyperbolic materials are anisotropic media for which the principal components of the permittivity tensor have opposite signs.[1] This class of natural and artificial materials enabled a series of major advances in nanophotonics, including but not limited to subdiffraction imaging,[2,3] negative refraction,[4] and emission energy control.[5] Metamaterials[1,3,4,6,7] and metasur-faces[8–10] comprised of artificially fabricated/stacked nanostruc-tures are most commonly employed to realize either 3D or 2D hyperbolic dispersions. In parallel with advances in metamate-rials, 3D and 2D hyperbolicity has been theoretically predicted

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scattering amplitude s(ω). In Figure 1, one can recognize bright fringes close to the sidewalls and slightly damped interference hot spots in the interior of the top surface. As was established in previous works,[11,15,23,27] these features originate from the standing wave interference between HPs launched by the AFM tip and those reflected by the hBN sidewalls. The HPs propa-gate as guided waves inside the hBN slab, undergoing multiple reflections at the top and bottom surfaces of the slab.[26] The period of polariton fringes on the hBN top surface equals to one half of the HP wavelength λHP/2.

In addition to the hallmarks of HPs, another series of polari-tonic fringes appear along the sidewalls of the hBN slab. These latter fringes exhibit the strongest amplitude close to hBN side-wall corners followed by slightly damped features away from the corner (Figure 1a). While these fringes display qualitatively similar interference patterns with those of the HPs, the side-wall polaritons reveal a distinct oscillation period. In Figure 2a, we plot the s-SNOM line traces taken along the red dashed and solid line cuts in Figure 1a. The polaritonic fringes in Figure 1a appear as oscillation peaks in the s-SNOM line traces (Figure 2a). Compared with the HPs (red dashed line), polaritons along the sidewall (red solid line) possess an oscillation period ≈20% shorter. Therefore, these polaritons can be identified as HSPs confined along the hBN slab sidewalls and experiencing multiple reflections between the top and bottom edges when propagating therein.[20] The formation of these propagating HSPs can be attributed to the 2D Type II hyperbolicity of the hBN sidewall.[9,18,20] The technique used to image HSPs is akin to that of HPs, except the reflectors of propagating polaritons (to form polariton standing wave fringes) are slab corners instead of slab sidewalls in hBN. The period of the polariton fringes on the sidewalls therefore equals to λHSP/2. Similar interference patterns originating from edge plasmon polaritons in graphene nanostructures have also been reported recently.[31,32]

Having established polaritonic edge modes, we now explore their propagation along the perimeter of diamond-shaped hBN flakes and also along the circumference of hBN disks (Figure 1a–d). Evident HSP fringes are observed around the hBN sidewall corners with an angle α in the range of 90°–180°. Additional s-SNOM images of the hBN slabs are provided in Section S1 (Supporting Information). We observe bright spots close to the hBN corner with α < 90° (Figure 1c,d). However, we do not attribute these features to polaritonic fringes in the following analysis, since the periodic oscillation fringes are missing. The amplitude of the HSP fringe oscillation decreases with increasing angle (Figures 1a,c,d and 2a) and vanishes in the hBN disk (Figure 1b) where no evident HSP features can be observed.

The evolution of HSP fringes as a function of the hBN corner angle α reveals the possibility of controlling and mani pulating the propagation of HSPs by tailoring α. HSPs launched with the s-SNOM tip propagate along the sidewall (green arrow in Figure 2c) and can be reflected, transmitted, and/or scattered into volume-confined HPs when reaching the corner. To inquire into the laws governing HSP propaga-tion, we first performed finite element simulations of HSPs at hBN flake’s corner with various apex angles α using COMSOL Multiphysics (see Section S3 in the Supporting Informa-tion for details). Simulations with representative corner angle α = 125° and 300° are shown in Figure 2c,d. In our simula-tions, the HSPs are launched from Port 1 (yellow arrow) and then travel along the sidewall I toward the corner. Note that we have filtered volume-confined HPs simultaneously launched from Port 1 in order to isolate the properties of HSPs. Upon reaching the corner, the propagating HSPs have three prin-cipal energy redistribution channels: 1) back reflection toward Port 1 along sidewall I (red arrow); 2) transmission through the corner and propagation toward Port 2 along sidewall II

Adv. Mater. 2018, 30, 1706358

Figure 1. Manipulation of hyperbolic surface polaritons in hBN nanodiamonds. a) Experimental setup. The AFM tip is illuminated by an IR beam from QCL. b–d) s-SNOM images of HSPs in hBN nanodiamonds with corner angle b) 180°, c) 135°, and d) 120° at ω = 1425 cm−1. Thickness of the hBN: 25 nm. Scale bar: 1 µm.



(cyan arrow); and 3) scattering at the corner with conversion into volume-confined HPs toward the interior of the hBN slab (gray arrows). In addition, the incident HSPs (yellow arrow) may also get scattered into free-space IR photons at the corner; however, in our simulation, this latter channel produced only negligible corrections. The reflection R, transmission T, and scattering S coefficients at the hBN corner can be defined as the ratio of the polariton energy in each channel over that of the incident HSPs: X = |EX|2/|Ein|2 (X = R, T, or S, respectively). By exploring the polariton energy in different ports, we can deter-mine the fraction of HSPs reflected, transmitted, and scattered into volume-confined HPs as a function of the corner angle α (Figure 2b). Since the imaging of HSPs in our experiments (Figure 1) relies on the reflection of incident HSPs at the hBN corner, the reflection coefficient R can be evaluated experimen-tally[33] (red squares in Figure 2b) by measuring the amplitude of the HSP fringes in our s-SNOM data (Figure 2a). The simu-lation results agree with our experimental data.

The polariton energy partition among these three chan-nels (R, T, and S) varies dramatically with the corner angle α (Figure 2b). At α ≈ 95°, nearly all incident HSPs are back reflected toward the Port 1, whereas the transmission T and scattering S are negligible. The reflection R decreases, while transmission T and scattering S grow with the increasing α. At α ≈ 125°, the scattering coefficient S becomes comparable to R and T, indicating a considerable portion of HSPs scattered at the corner and converted into HPs in the hBN interior (see also Figure 2c). At α = 180°, R and S both vanish such that all the

HSPs get transmitted, as expected in the absence of the corner. As α is further increased, R and T reveal the opposite trends, whereas the scattering of HSPs (S) increases significantly and exceeds 60% at the angle α = 300°. In the latter regime, HPs scattered from the HSPs can be clearly observed (Figure 2d). The evolution of R, T, and S stems from the fact that polariton energy distribution in these three channels is determined from the modal overlap integral[34,35] between the incident, reflected and transmitted HSPs, which strongly depends on the corner angle α.

Similar to previous studies of HPs in hBN,[11] the manipula-tion of HSPs can also be accomplished by changing the hBN slab thickness: a consequence of the waveguiding nature of propagating HSPs. The hBN thickness can be easily altered via mechanical exfoliation, since the hBN layers are bonded together via weak van der Waals forces.[22] At a representative IR frequency ω = 1425 cm−1, we demonstrate manipulation of the HSP wavelength by varying the thickness of the hBN slab (Figure 3). The HSP wavelength scales with the hBN thickness following a nearly linear law (red squares and solid line for data and simulation, respectively), consistent with the waveguided nature of HSPs. At this frequency, the HPs also obey a similar scaling law (red dots and dashed line), as reported before.[11]

We finally highlight the potential of steering HSPs by engi-neering the geometry of an hBN slab (Figure 4). Propagation steering of HPs and other polaritonic modes seem arduous without detailed design of the local dielectric environment in sophisticated structures,[36–38] since these polaritons can

Adv. Mater. 2018, 30, 1706358

Figure 2. Controlled reflection, transmission, and scattering of HSPs at the hBN corners. a) s-SNOM line profiles of HSPs at hBN slab corners with various corner angles. b) Simulated reflected (R), transmitted (T), and scattered to HPs (S) fractions of HSPs at 25 nm thick hBN flake’s corner as a function of the corner angle α. Red squares indicate the experimental data of the HSP reflection extracted from s-SNOM images at IR frequency ω = 1425 cm−1. Measurement error bars with a size larger than that of the data point are indicated. c,d) Simulated spatial distributions of |Re |zE of the fundamental HSPs mode as it reaches the hBN flake’s corners with apex angles c) α = 127° and d) α = 300°.



propagate along any direction in the basal plane (k//,HP, blue arrows) once they are launched. The surface-confined nature of HSPs restricts the propagation (k//,HSP, red arrows) along any chosen direction of the hBN sidewalls. In a proof-of-concept implementation, we engineered the hBN structure in Figure 4, observing HSP fringes not only along the straight sidewalls but also along the circumference of the fabricated semicircles. The propagation trace of HSPs can switch between a straight line and a designed curve (red arrows), following the engineered hBN sidewall orientation. Through propagation steering of polari-tons, HSPs can travel around the semicircle region (region A). Based on this concept, we envision the possibility of designing proper phase accumulation to realize a polaritonic cloak[39]: an important element of transformational polaritonics.[36,37]

The results presented in Figures 1–4 demonstrate the manip-ulation and steering of HSPs in hBN. The sidewall-confined nature of HSPs enables an efficient control of polariton

reflection, transmission, scattering, and propagation through tailoring the geometry of hBN nanostructures. The methodology that we utilized to manipulate and steer HSPs in hBN can be directly extended to surface polaritons propagating in hyperbolic metamaterials, metasurfaces,[9,16] and other van der Waals mate-rials, including black phosphorus[17] and topological insulators.[40] We envision future efforts directed toward hybridizing[41,42] HSPs with other polaritonic modes, such as plasmons in gra-phene and black phosphorus, for the realization of dynamically tunable hyperbolic metasurfaces and polaritonic flat optics.[16,43] The propagation steering demonstrated in Figure 4 may provide advantages for the implementation of transformation optics/polaritonics[36,37] over alternative platforms, by employing polari-tons propagating along the sidewall of natural or artificial hyper-bolic materials to engineer advanced waveguiding functionalities.

Experimental SectionExperimental Setup: The IR nanoimaging experiments introduced in

the main text were performed using a scattering-type scanning near-field optical microscope (s-SNOM). The s-SNOM is a commercial system (www.neaspec.com) based on a tapping-mode AFM. In the experiments, a commercial AFM tip (tip radius ≈10 nm) with a PtIr5 coating was used. The AFM tip was illuminated by monochromatic quantum cascade lasers (QCLs) (www.daylightsolutions.com) covering a frequency range of 900–2300 cm–1 in the mid-IR. The s-SNOM nanoimages were recorded by a pseudoheterodyne interferometric detection module with an AFM tapping frequency 280 kHz and tapping amplitude around 70 nm. In order to subtract background signal, the s-SNOM output signal was demodulated at the third harmonics of the tapping frequency.

Sample Fabrication: hBN crystals were mechanically exfoliated from bulk samples and deposited onto Si wafers capped with 285 nm thick SiO2. Oxygen plasma etching was utilized to pattern the hBN slabs into anticipated geometries.

Supporting InformationSupporting Information is available from the Wiley Online Library or from the author.

AcknowledgementsWork at UCSD and Columbia on optical phenomena in vdW materials was supported by the Gordon and Betty Moore Foundation’s EPiQS Initiative through grant no. GBMF4533. Research at UCSD and Columbia on metamaterials and development of nano-IR instrumentation was supported by AFOSRFA9550-15-1-0478 and ONR N00014-15-1-2671. P.J.-H. acknowledges support from AFOSR grant no. FA9550-16-1-0382. Research at UT Austin was supported by the AFOSR MURI grant no. FA9550-17-1-0002.

Conflict of InterestThe authors declare no conflict of interest.

Keywordshexagonal boron nitride, hyperbolic surface polaritons, polariton steering

Received: November 2, 2017Revised: January 22, 2018

Published online: March 13, 2018

Adv. Mater. 2018, 30, 1706358

Figure 3. Thickness dependence of HSPs and HPs in hBN. Dots and squares, experimental data extracted from s-SNOM images for HPs and HSPs in hBN, respectively. Dotted and solid lines, simulated thickness dependence for HPs and HSPs, respectively. IR frequency ω = 1425 cm−1.

Figure 4. Propagation steering of HSPs in hBN. s-SNOM image of HSPs in hBN with engineered geometry at ω = 1410 cm−1. Blue and red arrows indicate the propagation direction of HPs and HSPs in hBN, respectively. Scale bar: 1 µm.

http://www.neaspec.com

http://www.daylightsolutions.com



Adv. Mater. 2018, 30, 1706358

[1] A. Poddubny, I. Iorsh, P. Belov, Y. Kivshar, Nat. Photonics 2013, 7, 948.

[2] Z. Jacob, L. V. Alekseyev, E. Narimanov, Opt. Express 2006, 14, 8247.

[3] Z. Liu, H. Lee, Y. Xiong, C. Sun, X. Zhang, Science 2007, 315, 1686.[4] J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy,

X. Zhang, Science 2008, 321, 930.[5] D. Lu, J. J. Kan, E. E. Fullerton, Z. Liu, Nat. Nanotechnol. 2014, 9,

48.[6] D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, P. Rye, Appl. Phys.

Lett. 2004, 84, 2244.[7] A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz,

D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, C. Gmachl, Nat. Mater. 2007, 6, 946.

[8] A. V. Kildishev, A. Boltasseva, V. M. Shalaev, Science 2013, 339, 1232009.

[9] J. S. Gomez-Diaz, M. Tymchenko, A. Alù, Phys. Rev. Lett. 2015, 114, 233901.

[10] A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. de Leon, M. D. Lukin, H. Park, Nature 2015, 522, 192.

[11] S. Dai, Z. Fei, Q. Ma, A. S. Rodin, M. Wagner, A. S. McLeod, M. K. Liu, W. Gannett, W. Regan, K. Watanabe, T. Taniguchi, M. Thiemens, G. Dominguez, A. H. C. Neto, A. Zettl, F. Keilmann, P. Jarillo-Herrero, M. M. Fogler, D. N. Basov, Science 2014, 343, 1125.

[12] J. D. Caldwell, A. V. Kretinin, Y. Chen, V. Giannini, M. M. Fogler, Y. Francescato, C. T. Ellis, J. G. Tischler, C. R. Woods, A. J. Giles, M. Hong, K. Watanabe, T. Taniguchi, S. A. Maier, K. S. Novoselov, Nat. Commun. 2014, 5, 5221.

[13] J. Sun, N. M. Litchinitser, J. Zhou, ACS Photonics 2014, 1, 293.[14] A. Kumar, T. Low, K. H. Fung, P. Avouris, N. X. Fang, Nano Lett.

2015, 15, 3172.[15] Z. Shi, H. A. Bechtel, S. Berweger, Y. Sun, B. Zeng, C. Jin, H. Chang,

M. C. Martin, M. B. Raschke, F. Wang, ACS Photonics 2015, 2, 790.

[16] J. S. Gomez-Diaz, A. Alù, ACS Photonics 2016, 3, 2211.[17] A. Nemilentsau, T. Low, G. Hanson, Phys. Rev. Lett. 2016, 116,

066804.[18] N. Talebi, C. Ozsoy-Keskinbora, H. M. Benia, K. Kern, C. T. Koch,

P. A. van Aken, ACS Nano 2016, 10, 6988.[19] A. M. Dubrovkin, G. Adamo, J. Yin, L. Wang, C. Soci, Q. J. Wang,

N. I. Zheludev, Adv. Opt. Mater. 2017, 5, 1600768.[20] P. Li, I. Dolado, F. J. Alfaro-Mozaz, A. Y. Nikitin, F. Casanova,

L. E. Hueso, S. Vélez, R. Hillenbrand, Nano Lett. 2017, 17, 228.[21] D. N. Basov, M. M. Fogler, F. J. García de Abajo, Science 2016, 354,

195.[22] K. S. Novoselov, D. Jiang, F. Schedin, T. J. Booth, V. V. Khotkevich,

S. V. Morozov, A. K. Geim, Proc. Natl. Acad. Sci. USA 2005, 102, 10451.[23] E. Yoxall, M. Schnell, A. Y. Nikitin, O. Txoperena, A. Woessner,

M. B. Lundeberg, F. Casanova, L. E. Hueso, F. H. L. Koppens, R. Hillenbrand, Nat. Photonics 2015, 9, 674.

[24] A. J. Giles, S. Dai, O. J. Glembocki, A. V. Kretinin, Z. Sun, C. T. Ellis, J. G. Tischler, T. Taniguchi, K. Watanabe, M. M. Fogler, K. S. Novoselov, D. N. Basov, J. D. Caldwell, Nano Lett. 2016, 16, 3858.

[25] L. Novotny, B. Hecht, Principles of Nano-Optics, Cambridge University Press, Cambridge 2006.

[26] S. Dai, Q. Ma, T. Andersen, A. S. McLeod, Z. Fei, M. K. Liu, M. Wagner, K. Watanabe, T. Taniguchi, M. Thiemens, F. Keilmann, P. Jarillo-Herrero, M. M. Fogler, D. N. Basov, Nat. Commun. 2015, 6, 6963.

[27] P. Li, M. Lewin, A. V. Kretinin, J. D. Caldwell, K. S. Novoselov, T. Taniguchi, K. Watanabe, F. Gaussmann, T. Taubner, Nat. Commun. 2015, 6, 7507.

[28] X. G. Xu, B. G. Ghamsari, J.-H. Jiang, L. Gilburd, G. O. Andreev, C. Zhi, Y. Bando, D. Golberg, P. Berini, G. C. Walker, Nat. Commun. 2014, 5, 4782.

[29] A. J. Giles, S. Dai, I. Vurgaftman, T. Hoffman, S. Liu, L. Lindsay, C. T. Ellis, N. Assefa, I. Chatzakis, T. L. Reinecke, J. G. Tischler, M. M. Fogler, J. H. Edgar, D. N. Basov, J. D. Caldwell, Nat. Mater. 2018, 17, 134.

[30] S. Dai, Q. Ma, Y. Yang, J. Rosenfeld, M. D. Goldflam, A. McLeod, Z. Sun, T. I. Andersen, Z. Fei, M. Liu, Y. Shao, K. Watanabe, T. Taniguchi, M. Thiemens, F. Keilmann, P. Jarillo-Herrero, M. M. Fogler, D. N. Basov, Nano Lett. 2017, 17, 5285.

[31] Z. Fei, M. D. Goldflam, J. S. Wu, S. Dai, M. Wagner, A. S. McLeod, M. K. Liu, K. W. Post, S. Zhu, G. C. A. M. Janssen, M. M. Fogler, D. N. Basov, Nano Lett. 2015, 15, 8271.

[32] A. Y. Nikitin, P. Alonso González, S. Vélez, S. Mastel, A. Centeno, A. Pesquera, A. Zurutuza, F. Casanova, L. E. Hueso, F. H. L. Koppens, R. Hillenbrand, Nat. Photonics 2016, 10, 239.

[33] L. Jiang, Z. Shi, B. Zeng, S. Wang, J.-H. Kang, T. Joshi, C. Jin, L. Ju, J. Kim, T. Lyu, Y.-R. Shen, M. Crommie, H.-J. Gao, F. Wang, Nat. Mater. 2016, 15, 840.

[34] W.-P. Huang, J. Opt. Soc. Am. A 1994, 11, 963.[35] G. Roelkens, D. V. Thourhout, R. Baets, Opt. Lett. 2007, 32, 1495.[36] A. Vakil, N. Engheta, Science 2011, 332, 1291.[37] M. Kadic, S. Guenneau, S. Enoch, A. Huidobro Paloma,

L. Martín-Moreno, J. García-Vidal Francisco, J. Renger, R. Quidant, Nanophotonics 2012, 1, 51.

[38] A. Woessner, Y. Gao, I. Torre, M. B. Lundeberg, C. Tan, K. Watanabe, T. Taniguchi, R. Hillenbrand, J. Hone, M. Polini, F. H. L. Koppens, Nat. Photonics 2017, 11, 421.

[39] J. B. Pendry, A. Aubry, D. R. Smith, S. A. Maier, Science 2012, 337, 549.

[40] J.-S. Wu, D. N. Basov, M. M. Fogler, Phys. Rev. B 2015, 92, 205430.[41] S. Dai, Q. Ma, M. K. Liu, T. Andersen, Z. Fei, M. D. Goldflam,

M. Wagner, K. Watanabe, T. Taniguchi, M. Thiemens, F. Keilmann, G. C. A. M. Janssen, S. E. Zhu, P. Jarillo-Herrero, M. M. Fogler, D. N. Basov, Nat. Nanotechnol. 2015, 10, 682.

[42] V. W. Brar, M. S. Jang, M. Sherrott, S. Kim, J. J. Lopez, L. B. Kim, M. Choi, H. Atwater, Nano Lett. 2014, 14, 3876.

[43] N. Yu, F. Capasso, Nat. Mater. 2014, 13, 139.

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Supporting Information

for Adv. Mater., DOI: 10.1002/adma.201706358

Manipulation and Steering of Hyperbolic Surface Polaritons inHexagonal Boron Nitride

Siyuan Dai, Mykhailo Tymchenko, Yafang Yang, Qiong Ma,Marta Pita-Vidal, Kenji Watanabe, Takashi Taniguchi, PabloJarillo-Herrero, Michael M. Fogler, Andrea Alù,* and DimitriN. Basov*

1

Supporting Information

Manipulation and steering of hyperbolic surface polaritons in hexagonal boron nitride

Siyuan Dai, Mykhailo Tymchenko, Yafang Yang, Qiong Ma, Marta Pita-Vidal, Kenji

Watanabe, Takashi Taniguchi, Pablo Jarillo-Herrero, Michael M. Fogler, Andrea Alù*,

Dimitri N. Basov*

1. Nano-imaging of hBN diamond with other corner angles

In Figure S1, we provide infrared (IR) nano-images of hyperbolic surface polaritons

(HSPs) in hBN with corner angle 105o (Figure S1a) and 150

o (Figure S2a) obtained by our

scattering type scanning near-field optical microscope (s-SNOM). These data show similar

features as those in Figure 1 of the main text and follow the trend of the evolution of the

reflection coefficient R. Line traces of the HSPs in Figure S1 are provided in Figure 2a of the

main text.

Figure S1. Additional s-SNOM images of hyperbolic surface polaritons in hBN nano

diamonds. s-SNOM amplitude images of hBN nano diamond with corner angle 105o (a) and

150o (b) at = 1425 cm

-1. Thickness of the hBN: 25 nm. Scale bar: 1 m.

2. s-SNOM and atomic force microscope (AFM) data of hBN with different thickness

At a representative IR frequency = 1425cm-1

, we characterized hBN crystals with

different thickness, the simultaneously recorded s-SNOM line traces and AFM topography are

2

presented in Figure S2a and Figure S2b, respectively. HSPs wavelength and the hBN

thickness are extracted from Figure S2 and are shown in Figure 3 of the main text.

Figure S2. s-SNOM and AFM line traces from hBN crystals with different thickness at =

1425cm-1

.

3. Numerical modeling of polaritons in hBN

3

Numerical analysis of HSPs propagation, reflection, and transmission through the hBN

corner was performed using COMSOL Multiphysics. For this purpose, we designed a fully

three-dimensional (3D) model of an hBN layer residing on top of the thick SiO2 substrate. The

two straight edges (cyan lines in Figure 2c and 2d) comprising the hBN corner had an equal

length l = 2µm. The two numerical waveguide ports were placed at the outer edges of the

model, and configured to excite and absorb only the lowest-order HSP with an effective index

n = n’ – jn” (here, the prime and double primes denote the real and imaginary part,

respectively, and we adopt the ejt

sign convention). In order to obtain the energy reflection R

and transmission T at the corner, we record the scattering parameters of HSPs at Port 1 and

Port 2, Sij (here i and j denote the port indexes). The scattering parameters[1]

at the corner,

(i and j are port numbers), are related to the scattering parameters at the ports as

, where k0 = / c is the free-space wavenumber, and the exponential factor

accounts for the accumulated propagation phase and losses. The reflection, transmission, and

scattering coefficients at the corner can be computed as ,

and

, respectively. Note that the length l and other dimensions of the model were chosen to

be large enough to ensure that bulk hyperbolic polaritons (HPs), as well as higher-order HSPs

scattered from the incident HSPs at the hBN corner, sufficiently decay before reaching Port 1,

Port 2 and the outer boundaries of the model. Finally, Figures 2c and 2d were obtained by

post-processing the simulated spatial field distribution at 70nmz and filling the domain

outside of the model with zero field.

References

[1] R. E. Collin, Foundations for Microwave Engineering, 2nd edition, Wiley, New York,

2001.

manipulation and steering of hyperbolic surface polaritons in...

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